The Hamiltonian H = xp and classification of osp ( 1 | 2 ) representations

The quantization of the simple one-dimensional Hamiltonian H = xp is of interest for its mathematical properties rather than for its physical relevance. In fact, the Berry-Keating conjecture speculates that a proper quantization of H = xp could yield a relation with the Riemann hypothesis. Motivated by this, we study the so-called Wigner quantization of H = xp, which relates the problem to representations of the Lie superalgebra osp(1|2). In order to know how the relevant operators act in representation spaces of osp(1|2), we study all unitary, irreducible ∗-representations of this Lie superalgebra. Such a classification has already been made by J. W. B. Hughes, but we reexamine this classification using elementary arguments.